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# Гармонический анализ на плоскости Лобачевского-Галилея.

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```Âåñòíèê ÒÃÓ, ò. 16, âûï. 6, ÷. 2, 2011
MSC 43A85
Harmonic analysis on the LobachevskyGalilei
plane 1
c
Yu. V. Dunin
Derzhavin Tambov State University, Tambov, Russia
Harmonic analysis on the LobachevskyGalilei plane is constructed
Keywords: Lobachevsky plane, dual numbers, Poisson and Fourier transforms, spherical
functions, Plancherel formula
 1. LobachevskyGalilei plane
Let Λ be an algebra over R of dimension 2 consisting of elements z = x + iy ,
x, y ∈ R with relation i2 = 0 (the algebra of dual numbers). It is not a eld: pure
imaginary numbers iy are zero divisors. For z = x + iy , the conjugate number is
z = x − iy .
The LobachevskyGalilei plane L is a domain on the plane Λ, dened by zz < 1.
It is a vertical strip bounded by lines x = ±1. Denote the line x = 1 by Γ. The group
G of translations of the LobachevskyGalilei plane L consists of linear-fractional
transformations
z 7→ z · g =
az + b
, aa − bb = 1, a, b ∈ Λ.
bz + a
It preserves the measure
dσ(z) =
Matrices
g=
a b
b a
(1.1)
dx dy
.
(1 − x2 )2
, aa − bb = 1,
occuring in (1.1), form the group SU(1, 1; Λ). Denote a = α + ip, b = β + iq . The
condition aa − bb = 1 is equivalent to α2 − β 2 = 1, so that α2 > 1, therefore α > 1 or
α 6 −1. Thus, the group SU(1, 1; Λ) consists of two connected parts. The connected
component of the identity is isomorphic just to the group G. This component consists
1 Supported
by the Russian Foundation for Basic Research (RFBR): grant 09-01-00325-à, Sci.
Progr. "Development of Scientic Potential of Higher School": project 1.1.2/9191, Fed. Object
Progr. 14.740.11.0349 and Templan 1.5.07
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Âåñòíèê ÒÃÓ, ò. 16, âûï. 6, ÷. 2, 2011
of matrices g with α > 1. For this component we retain the notation G. We can
write parameters α and β of the matrix g ∈ G as α = ch t è β = sh t, where t ∈ R.
Therefore, any matrix g ∈ G can be written as follows
g = g(t) + ic(p, q),
where
g(t) =
ch t sh t
sh t ch t
(1.2)
, c(p, q) =
p
q
−q −p
.
The stabilizer of the point z = 0 is the subgroup K consisting of diagonal
matrices:
1 + ip
0
k=
,
(1.3)
0
1 − ip
so that
L = G/K.
Let (f, h)L be the inner product in the space L2 (L, dσ):
Z
(f, h)L =
f (z) h(z) dσ(z).
L
The quasiregular representation U of G acts on this space: (U (g)f )(z) = f (z · g).
 2. Representations of the group G
In this section we describe two series of representations of the group G induced by
characters (one-dimensional representations) of "parabolic" subgroups P0 è P∞ . The
rst one P0 is the stabilizer of the point γ0 = 1 in Γ, the second one P∞ is obtained
by a limit passage from the stabilizer of the point γy = 1 + iy when y → ∞.
The subgroup P0 consists of matrices (1.2) with p = q , i. e. matrices h(t, q) =
g(t) + ic(q, q). Its character ωλ is dened by a complex number λ:
ωλ (h) = eλt = (α + β)λ .
The set G/P0 can be identied with the line Γ: to a point γy one assigns the diagonal
matrix (1.3) with p = y/2. The representation Tλ of G induced by ωλ acts in functions
ϕ(γ) in D(Γ) by
Tλ (g)ϕ(γ) = ϕ (γ · g) (α + β)λ , λ ∈ C.
The stabilizer Py of the point γy consists of matrices (1.2) such that p = q +y sht.
The subgroups Py and P0 are isomoephic, so representations induced by characters
these subgroups are equivalent.
Let us nd the limit P∞ of Py when y → ∞. We consider parameters t and q
depending on y : t = ty , q = qy , in such way that the following limits exist: lim ty = t,
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Âåñòíèê ÒÃÓ, ò. 16, âûï. 6, ÷. 2, 2011
lim qy = q , lim(qy + y shty ) = p. It gives t = 0. Therefore, the subgroup P∞ consists
of matrices
h = h(u, v) = E + ic(u, v),
(2.1)
where E is the identity matrix of the second order. This subgroup is a commutative
normal subgroup of G, isomorphic to R2 , so that its characters are
ωλ,µ (h) = eλu eµv ,
where λ, µ ∈ C and h is the matrix (2.1). Any matrix (1.2) can be written as
g = h(u, v) g(t),
(2.2)
where
h(u, v) = E + ic(p, q)g(t)−1 ,
so that
u = p ch t − q sh t, v = −p sh t + q ch t.
So we can identify G/P∞ with the subgroup of matrices g(t) and hence with R. The
representation Tλ,µ of G induced by ωλ,µ acts in functions ϕ(s) in D(R) by:
Tλ,µ (g)ϕ(s) = ϕ(e
s) ωλ,µ (e
h),
where se and e
h are obtained if we decompose g(s) g in accordance with (2.2): g(s) g =
e
h g(e
s). Let us write g also as (2.2) then
g(s) g = {E + ig(s)c(u, v)g(s)−1 } · g(s + t),
so that se = s + t, e
h(u, v) = E + i c(e
u, ve), where
c(e
u, ve) = g(s)c(u, v)g(s)−1
= g(s)c(p, q)g(s + t)−1 ,
therefore,
u
e = p ch(2s + t) − q sh(2s + t),
ve = −p sh(2s + t) + q ch(2s + t).
(2.3)
(2.4)
Finally for g = g(t) + ic(p, q) we get
(Tλ,µ (g) ϕ) (s) = ϕ(s + t) eλeu+µev ,
(2.5)
with u
e and ve given by (2.3) and (2.4).
A Hermitian form (the inner product from L2 (R, ds))
Z ∞
hψ, ϕi =
ψ(s)ϕ(s) ds
−∞
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Âåñòíèê ÒÃÓ, ò. 16, âûï. 6, ÷. 2, 2011
is invariant with respect to the pair (Tλ,µ , T−λ,−µ ), i. e.
hTλ,µ (g)ψ, ϕi = hψ, T−λ,−µ (g −1 )ϕi,
(2.6)
so that Tλ,µ is unitarizable for pure imaginary λ, µ.
Using (2.6) we can extend Tλ,µ to the space D0 (R) of distributions ψ on R.
 3. Poisson and Fourier transforms, spherical functions
We need the second series from  2.
Theorem 3.1
provided that
K -invariants in D0 (R) under Tλ,µ exists
−1 < r < 1; if so let us set r = th 2τ , τ ∈ R, so
A non-trivial space of
λ = rµ,
where
that
λ = th 2τ · µ, τ ∈ R.
This space is one-dimensional, a basis function
θ
(3.1)
is the delta function:
θ(s) = δ(s − τ ).
Proof. Let a function θ(s) is K -invariant under Tλ,µ . By (1.3) and (2.5), (2.3), (2.4)
it means:
ep(λ ch 2s−µ sh 2s) θ(s) = θ(s)
for all p ∈ R. This is equivalent to a condition that is obtained by dierentiation
with respect to p at zero: (λ ch 2s − µ sh 2s) θ(s) = 0, or (λ − µ th 2s) θ(s) = 0. The
factor in front of θ(s) has to vanish at some point s = τ . It gives (3.1).
The representation Tλ,µ with condition (3.1) is equivalent to that with λ = 0.
Indeed, the translation operator C : (Cϕ)(s) = ϕ(s + τ ), intertwines Tλ,µ with T0,ν ,
where ν = µ/ch 2τ . So we can take λ = 0 from the beginning. Then Theorem 3.1
claims that the representation T0,µ , µ ∈ C, has a K -invariant θ(s) = δ(s) unique up
to a factor.
The K -invariant θ(s) = δ(s) gives rise to a Poisson kernel
Pµ (z, s) = T0,µ (g −1 )θ (s), z ∈ L, s ∈ R,
where g is an element in G moving 0 to z , for example, the matrix
1
1 z
, z = x + iy.
gz = √
1 − x2 z 1
We get
y
Pµ (z, s) = δ(ξ − s) exp µ
1 − x2
2
= (1 − x ) δ(x − c) exp µ
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y
1 − x2
,
Âåñòíèê ÒÃÓ, ò. 16, âûï. 6, ÷. 2, 2011
where x = th ξ , c = th s. This kernel gives rise to two transforms: the Poisson
∞
transform Pµ : D(R) → C (L) and the Fourier transform Fµ : D(L) → D(R)
dened respectively by:
Z ∞
y
Pµ (z, s) ϕ(s) ds = ϕ(ξ) exp µ
(Pµ ϕ) (z) =
,
(3.2)
1 − x2
−∞
Z
(Fµ f ) (s) =
Pµ (z, s) f (z) dσ(z)
L
Z ∞
1
y
=
dy, c = th s.
f (c + iy) exp µ
1 − c2 −∞
1 − c2
They intertwine T0,−µ with U and U with T0,µ , respectively. These transforms are
conjugate to each other:
(Pµ ϕ, f )L = hϕ, Fµ f i.
The
spherical function
Ψµ is dened as the Poisson image of the K -invariant θ:
Ψµ = Pµ θ.
It follows from (3.2) that
Ψµ (z) = δ(x) eµy .
 4. Decomposition of the quasiregular representation
Theorem 4.1
The quasiregular representation
U
decomposes in the direct integral of representations
√
L2 (L, dσ)
of the group
G
T0,iρ , ρ ∈ R,
with multiplicity
in
i = −1 ∈ C, the complex number). Let us assign to a
f ∈ D(L) the family of its Fourier components Ôóðüå Fiρ f , ρ ∈ R. This
correspondence is G-equivariant. There are an inverse formula:
Z ∞
1
f=
P−iρ Fiρ f dρ,
(4.1)
2π −∞
one as follows (here
function
and the Plancherel formula:
1
(f, h)L =
2π
Therefore the map
f 7→ {Fiρ f }
Z
∞
hFiρ f, Fiρ hi dρ.
(4.2)
−∞
can be extended to the whole space
L2 (L, dσ).
Formulas (4.1), (4.2) are obtained from corresponding formulas for the classical
Fourier transform by the change ρ = (1 − x2 )η .
These formulas can be united by the formula decomposing the delta function
δ(z) concentrated at z = 0 into spherical functions:
Z ∞
1
δ(z) =
Ψiρ (z) dρ.
2π −∞
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